Combine turnips and water to cover in a saucepan. Bring to a boil; cook 20 minutes or until tender. Drain well; transfer to a large bowl, and mash with a potato masher.
Melt 1 tablespoon butter in a skillet over medium heat; add onion, and sauté 3 minutes or until tender.
Add onion, milk, 1 cup Cheddar cheese, salt, pepper, and remaining 3 tablespoons butter to mashed turnips, stirring to combine. Spoon into a lightly greased 11- x 7-inch baking dish. Sprinkle evenly with breadcrumbs and remaining 1/2 cup Cheddar cheese.
Bake at 350° for 30 minutes or until lightly browned.